Conics - Precalculus

Front

What is a conic section?

Back

A conic section is a curve obtained by intersecting a cone with a plane. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas.

Front

What is the standard form of a parabola that opens upwards?

Back

The standard form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Front

What are the characteristics of a circle?

Back

A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. The standard form is (x - h)² + (y - k)² = r².

Front

What is the equation of an ellipse in standard form?

Back

The standard form of an ellipse is (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis.

Front

What is the definition of a hyperbola?

Back

A hyperbola is a set of all points where the difference of the distances to two fixed points (foci) is constant. The standard form is (x - h)²/a² - (y - k)²/b² = 1.

Front

How do you find the focus of a parabola?

Back

For a parabola in the form y = a(x - h)² + k, the focus is located at (h, k + 1/(4a)).

Front

What are the vertices of the ellipse given by the equation x² + 9y² - 4x + 54y + 49 = 0?

Back

The vertices are (8, -3) and (-4, -3).

Front

What is the relationship between the foci and vertices of a hyperbola?

Back

For a hyperbola, the distance from the center to each focus (c) is greater than the distance from the center to each vertex (a). The relationship is c² = a² + b².

Front

What is the general form of a conic section?

Back

The general form of a conic section is Ax² + Bxy + Cy² + Dx + Ey + F = 0, where A, B, C, D, E, and F are constants.

Front

How can you determine if a conic is a hyperbola?

Back

A conic is a hyperbola if the discriminant B² - 4AC > 0 in the general form Ax² + Bxy + Cy² + Dx + Ey + F = 0.